LiNA A Graphical Matlab Tool for
Analyzing Intrinsic Noise in
Biochemical Reaction Networks
Tutorials and Theoretical Background
Ronny Straube and Axel von Kamp
Analysis and Redesign of Biological Networks Group
Max Planck Institute for Dynamics of Complex Technical Systems
Sandtorstrasse 1, 39106 Magdeburg, Germany
August 19, 2013
Contents
1. Installation and License Information
1.1.
1.2.
1.3.
1.4.
Availability . . . . .
Installation . . . . .
License Information .
Contact Information
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2.1. Noise Reduction Through Receptor Dimerization . . . . .
2.1.1. Setting up the Model . . . . . . . . . . . . . . . . .
2.1.2. Taking A First Look . . . . . . . . . . . . . . . . .
2.1.3. Volume Dependence of the Fluctuations . . . . . .
2.1.4. Fano Factor . . . . . . . . . . . . . . . . . . . . . .
2.1.5. Parameter Sets, Sessions and Figure Export . . . .
2.2. Ultrasensitivity in the Goldbeter-Koshland Model . . . . .
2.2.1. Analysis of Systems with Conservation Relations .
2.2.2. Correlation Analysis . . . . . . . . . . . . . . . . .
2.3. Bistability in Covalent Modication Cycles . . . . . . . . .
2.3.1. Model Setup . . . . . . . . . . . . . . . . . . . . .
2.3.2. Strategies for Dealing with Multiple Steady States
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2. Tutorials
3. Technical Remarks
3.1. Solving Eqs. (3.1) and (3.2) . . . . . . . . . . . . .
3.2. Rescan vs. All Steady States . . . . . . . . . . . . .
3.3. Data Management . . . . . . . . . . . . . . . . . .
3.4. Output at MATLAB Console and vanKampen.out
4. Theoretical Background
4.1.
4.2.
4.3.
4.4.
4.5.
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Reaction Networks and the Master Equation . . . . . . . .
The Linear Noise Approximation . . . . . . . . . . . . . .
Calculation of Stochastic Quantities . . . . . . . . . . . .
Dealing with Mass Conservation . . . . . . . . . . . . . .
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1. Noise Reduction Through Receptor Dimerization .
4.5.2. Receptor-Ligand Binding with Mass Conservation .
A. Derivation of the Linear Noise Approximation
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4
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32
36
A.1. Transformation of the Master Equation . . . . . . . . . . . . . . . . . . . . 36
2
Contents
~ t . . . . . . . . . . . . . . . . . . . 39
A.2. Eective Equations for ~x (t) and Π ξ,
A.3. Equations for hξi (t)i and σii0 (t) = hξi ξi0 i . . . . . . . . . . . . . . . . . . . 40
3
1. Installation and License Information
1.1. Availability
LiNA can be downloaded from
http://www.mpi-magdeburg.mpg.de/projects/LiNA
1.2. Installation
Unzip the LiNA.zip archive to your HOME directory or to any other directory of your
choice. This will create a directory called LiNA containing the program les. Example
networks described in the Tutorial Chapter (→ Chapter 2) can be found in the Examples
folder which is located in the LiNA directory.
1.3. License Information
LiNA is free software. It is published under the terms of the GNU General Public
License v3. A full copy of the license terms and conditions can be found in the le
named COPYING which is located in the LiNA directory.
1.4. Contact Information
Please send your comments, suggestions or bug reports to
[email protected]
4
2. Tutorials
If you are familiar with the linear noise approximation you may want to have a look
at the following Tutorials which cover all of the functionality implemented in LiNA. If
you don't know the linear noise approximation you may nd it useful to rst read the
background information in Chapter 4 and Appendix A before coming back.
Each tutorial highlights dierent aspects of LiNA's functionality. For example, in the
rst tutorial one learns how to setup a model, how to dene compounded parameters
and how to use the graphical user-interface for analyzing basic system properties. This
tutorial also covers data management tasks such as loading/saving sessions and exporting
gures. The second tutorial shows how LiNA handles systems with conservation relations
and how to analyze correlations. The last tutorial discusses the functionality provided
to analyze systems with multiple steady states. This tutorial refers to LiNA v1.
2.1. Noise Reduction Through Receptor Dimerization
In the rst tutorial we shall analyze how dimerization aects the steady state uctuations
of receptor molecules. To this end, we consider the reaction system
ks
∅ R
kd1
k+
R + R R2
R2
k−
kd2
→ ∅
(2.1)
(2.2)
(2.3)
which represents an extension of the model considered by Hayot and Jayaprakash [5].
In the absence of dimerization, synthesis and degradation of receptor monomers, as
described by Eq. (2.1), lead to Poissonian uctuations where the average number of
monomers is equal to the variance so that the Fano factor is 1. Hayot and Jayaprakash
showed that taking dimerization into account (Eq. 2.2), but neglecting dimer degradation (Eq. 2.3), the steady state uctuations for both, monomers and dimers, remain
Poissonian within the linear noise approximation [5]. In Section 4.5.1 we re-analyze
the system considered by Hayot and Jayaprakash, but for a nite lifetime of receptor
dimers (kd2 6= 0), which is shown to lead to sub-Poissonian uctuations for monomers
and dimers. Interestingly, these uctuations can be minimized either for monomers or
for dimers depending on the parameter regime. In the following, we will use LiNA to
visualize some of the theoretical results obtained in Section 4.5.1.
5
2. Tutorials
Figure 2.1.: Startup window of LiNA.
2.1.1. Setting up the Model
To start LiNA, startup MATLAB rst. If necessary change to the LiNA directory (see
Section 1.2). Then, type startlina at the MATLAB prompt and press ENTER. This
will bring up the Equations window (see Fig. 2.1) which allows you to enter new reactions, specify initial parameter values and to dene new parameters through parameter
substitutions. You can also load an existing model using the Load... button. In the
opening le selection window change to the Examples folder and select the le receptor_dimerization.mat after which the Equations window will look as in Fig. 2.2.
The Equation Editor
In the rst and in the third reaction the digit `0' is used to denote unspecied sources and
sinks. Irreversible reactions are entered using a single arrow (->), reversible reactions
using a double arrow (<->). New reactions can be added with the Add Reaction button.
Further information on this topic can be obtained by pressing the ? button in the upper
right corner of the Equation s window. Reactions can be deleted using the delete button
(red circle with the white cross) which is located to the left of each reaction eld.
For each reaction at least one parameter has to be specied; two for reversible reactions.
The parameter in the left column refers to the forward reaction (→) whereas that in the
right column refers to the backward reaction (←). Parameter names are predened, but
can be changed (only alphanumeric characters).
A Note on Units
The unit of a parameter depends on the order of the reaction and on the chosen unit
for the species concentration. The default unit for the species concentration is µM =
10−6 mol/liter. In that case, the value of the parameter ks , which denes the rate with
which monomers are synthesized (0 → R) , is expected to be entered in units of µM/time
where `time ' could be seconds, minutes or hours. The particular time unit is unimportant
6
2. Tutorials
Figure 2.2.: Model setup for the reaction system in Eqs. (2.1) - (2.3).
since, in its current form, LiNA only calculates noise characteristics under steady state
conditions which do not depend on the specic time scale. However, it is important
that all parameter values are given with respect to the same time scale. The rst
order rate constants kd1 (kd1), kd2 (kd2) and k − (km), which describe the degradation of
monomers, dimers and the dissociation of the dimer, respectively, have the unit 1/time
where the time unit has to be the same as that used for ks . Finally, the second-order rate
constant k + (kp), which describes the association of two monomers into one dimer, is
expected to be entered in units of 1/(µM · time ). If the species concentration is changed,
e.g. from µM to nM , the values for zeroth and second order rate constants (ks and k + ),
are re-interpreted accordingly, i.e. as nM/time for ks and 1/(nM · time ) for k + . The
values of rst order rate constants are not aected by such a change.
Why do I need to dene the reaction volume?
Calculation of the standard deviation and, hence, calculation of the coecient of variation
depends on the reaction volume (cf. Eq. 4.17). The default volume is set to V =
10−15 l = 1f l = 1µm3 which corresponds to the typical volume of a bacterial cell. In
such a small volume a concentration of x = 10nM means that there are approximately
n = V · NA · x ≈ 6 particles in the system (NA = 6.022 · 1023 particles/mol). This is the
regime where one can expect stochastic eects to become important.
Parameter Substitutions
In Subsection 4.5.1 we show that the steady states as well as the steady state uctuations
of the reaction system in Eqs. (2.1) - (2.3) only depend on certain parameter combinations
7
2. Tutorials
which can be chosen as (cf. Eq. 4.49)
Rs =
ks
,
kd1
α=
kd1
,
kd2
β=
kd2
,
k−
KD =
k−
.
k+
(2.4)
In order to compare the results, generated by LiNA, with the theoretical results from
Subsection 4.5.1 it is, thus, helpful to be able to dene the same parameters in LiNA.
This is the purpose of the Parameter Substitutions eld shown in Figure 2.2. Using
the expressions in Eq. (2.4) the parameters kd2 , kd1 , ks and k + have been replaced by
Rs (Rs), α (a), β (b) and KD (Kd) according to
kd2 = βk − ,
kd1 = αβk − ,
ks = Rs αβk − ,
k + = k − /KD .
The biological meaning of these parameters is immediately evident: Rs represents the
steady state level of monomers in the absence of dimerization, α diers from 1 if monomers
and dimers are degraded at dierent rates, β compares the time scales for the dimer
degradation and dimer dissociation, and KD represents the dimer dissociation constant.
2.1.2. Taking A First Look
If the model setup is nished the analysis can be started by pressing the Start button
(Fig. 2.2). This will bring up two new windows, denoted as Control and Plots (Fig. 2.3),
while the Equations windows becomes `frozen'. The rst quantity to be plotted is always
the Mean Values of all species as a function of the rst entry in the parameter list of
the Parameters eld (A), Kd in this case. The parameter list contains the newly dened
parameters (cf. Eq. 2.4) as well as k − (km) which has not been substituted. Note,
however, that neither the steady states nor any of the stochastic quantities depends
explicitly on k − . You can readily check this by choosing k − (km) from the Parameter
pull-down list in the X-Axis:Variable Parameter eld (B). Then you should see 2 straight
lines parallel to the x-axis.
2.1.3. Volume Dependence of the Fluctuations
To see how the reaction volume aects the magnitude of steady state uctuations due
to intrinsic reaction noise choose again Kd from the Parameter pull-down list in the
X-Axis:Variable Parameter eld (B) and select Mean with Errorbars from the pull-down
list in the Y-Axis eld (C). This should add some errorbars to the steady state curves.
The errorbars represent the standard deviation as dened in Section 4.3. Now, if the
reaction volume is decreased the average number of particles also decreases and particle
uctuations become larger. Conversely, if the reaction volume is increased particle uctuations become reduced. You can test this idea by changing the value of the reaction
volume: Select Vol. in the parameter list of the Parameters eld (A) and change the
value from 1e − 15 to 1e − 14 and, subsequently, to 1e − 16. Don't forget to press Apply
Changes for parameter changes to take eect.
8
2. Tutorials
Figure 2.3.:
Control window (left) and Plots window (right) of LiNA.
2.1.4. Fano Factor
If monomers and dimers are degraded at the same rate (α = 1) and if dimer degradation occurs at a much slower rate than dimer dissociation (β 1) the Fano factors of
monomers and dimers can be approximated by (cf. Eqs. 4.55 and 4.56)
F1 =
s
σ11
xs1 /KD
≈
1
−
xs1
(1 + 4xs1 /KD )2
F2 =
s
σ22
1 (4xs1 /KD )2
≈
1
−
xs2
4 (1 + 4xs1 /KD )2
where xs1 and xs2 are the steady state values of monomers (Eq. 4.47) and dimers
(Eq. 4.48), respectively. Hence, uctuations in both, monomers and dimers, are subPoissonian. However, whereas monomer uctuations, again, become Poissonian (F1 ≈ 1)
as xs1 KD /4 dimer uctuations are reduced and the Fano factor approaches F2 ≈ 3/4.
To reproduce these results with LiNA choose Rs (Rs) from the Parameter pull-down
list in the X-Axis:Variable Parameter eld (B), change the Range from 0.01 to 1000,
change to log scale and choose Fano Factor from the pull-down list in the Y-Axis eld
(C). This should produce two curves as in Figure 2.4. Convince yourself that changing
the binding anity KD (Kd) merely shifts these curves to the left (if Kd is decreased)
or to the right (if Kd is increased). Also, observe that increasing β while keeping α = 1
xed successively shifts the Fano factor curve for monomers below that for dimers, so
that monomer uctuations are reduced. Now, keep β = 0.01 xed and reduce α so that
α 1. This reduces the Fano factor for monomers at intermediate values of Rs where
dimer uctuations are still approximately Poissonian (region A in Fig. 2.5). At large
9
2. Tutorials
Figure 2.4.: Fano factors for monomers (R) and dimers (R2) as a function of Rs = ks /kd1
for KD = 1µM , α = 1 and β = 0.01.
values of Rs the situation is reversed (region B in Fig. 2.5). Hence, if dimers are degraded
at a much higher rate than monomers it depends on the value of Rs whether uctuations
in monomers or dimers are reduced.
2.1.5. Parameter Sets, Sessions and Figure Export
As you keep playing around with dierent parameter combinations LiNA will add a
corresponding Parameter set for each distinct parameter combination. The number of
available parameter sets is shown in round brackets in the Parameters eld. To choose
a particular parameter set, simply change the number in the Parameter Set: eld and
press Enter. When you change the variables along the x-axis or the y-axis or if you
change parameters LiNA adds a new gure to the plot list which can be accessed from
the pull-down menu in the Plots window (D in Fig. 2.3).
The plots are labeled according to the format
parameter on the x-axis: quantity on the y-axis {parameter set}.
For example, in Fig. 2.5 the plot shows the Fano factor as a function of Rs using
parameter set 3. Note that parameter sets cannot be deleted. You can, however, delete
a plot by pressing the Delete button next to the pull-down menu in the Plots window.
If you also want to delete the calculations associated with a plot you have to use the
delete button (red circle with a white cross) next to the Parameter pull-down list in the
X-Axis: Variable Parameter eld (B in Fig. 2.3).
10
2. Tutorials
Figure 2.5.: Fano factors for monomers (R) and dimers (R2) as a function of Rs = ks /kd1
for KD = 1µM , α = 0.001 and β = 0.01.
If you wish to continue the analysis at a later time you can save the current session
including parameter sets and plots using the save button (disk symbol right above the
Y-Axis eld) in the Control window. Saved sessions can be loaded using the Resume
Session ... button from the startup window of LiNA (Fig. 2.1).
If you want to save a gure for further processing in MATLAB you can either directly
save the plot as a MATLAB gure le (using the disk symbol in the Plots window) or
you copy the contents of the current plot (using the copy symbol labeled by E in Fig.
2.3) which creates a new MATLAB gure so that you can add comments, change fonts
or convert the gure to another format before saving. Note that the plots in a copied
gure will not receive any further updates as you continue with your analysis.
2.2. Ultrasensitivity in the Goldbeter-Koshland Model
The Goldbeter-Koshland model is a classical model for the description of covalent modication systems [3] where a substrate molecule is interconverted between an unmodied
(S ) and a modied (S ∗ ) form by a pair of opposing converter enzymes. In the case of
phosphorylation / dephosphorylation cycles the converter enzymes are called kinase (K )
and phosphatase (P ). In the simplest case the enzyme-catalyzed reactions are modeled
as irreversible Michaelis-Menten type reactions of the form
k1+
k1
∗
∗
k2+
k
S + P S ∗ -P →2 S + P.
S + K S -K → S + K,
k1−
k2−
11
(2.5)
2. Tutorials
Figure 2.6.: Model setup for the reaction system in Eq. 2.5.
Here, we have assumed that the reactions occur under in vitro conditions so that
synthesis and degradation of substrate and converter enzymes can be neglected. Hence,
there are three conservation relations
[S] + [S ∗ ] + [S -K] + [S ∗ -P ] = ST
[K] + [S -K] = KT
[P ] + [S ∗ -P ] = PT
corresponding to the conservation of the total amounts of substrate (ST ), kinase (KT )
and phosphatase (PT ).
If the total substrate concentration is much larger than that of either converter enzyme
(ST max (KT , PT )) one can describe the dynamics of the modied form of the substrate
by the eective equation [3]
d [S ∗ ]
ST − [S ∗ ]
[S ∗ ]
≈ k1
−
k
2
dt
K1 + ST − [S ∗ ]
K2 + [S ∗ ]
where K1 = (k1 +k1− )/k1+ and K2 = (k2 +k2− )/k2+ denote the Michaelis-Menten constants
of the kinase and the phosphatase, respectively. A hallmark of the Goldbeter-Koshland
model is that, under steady state conditions d [S ∗ ] /dt = 0, it can generate highly sigmoidal response curves if both converter enzymes operate in saturation (Fig. 2.7), i.e. if
max(K1 , K2 ) ST . This phenomenon is called zero-order ultrasensitivity.
2.2.1. Analysis of Systems with Conservation Relations
To setup the model start LiNA from the MATLAB prompt with startlina and enter
reactions and parameters as shown in Fig. 2.6. Alternatively, you may load the le Gold-
12
2. Tutorials
beter_Koshland_model.mat from the Examples folder. Note that the `on' rate constants
k1+ and k2+ have been replaced by the associated Michaelis-Menten constants according
to ki+ = (ki + ki− )/Ki (i = 1, 2) where Ki ↔ Kmi. After starting the calculation (press
Start) you will see three new parameters (C_K, C_Sp and C_P) in the parameter list
of the Parameters eld (Fig. 2.7). LiNA automatically adds a new parameter for each
conservation relation it detects in the network. Hence, C_K, C_Sp and C_P correspond to the total concentrations of kinase, substrate and phosphatase. LiNA displays
the corresponding conservation relation in the X-Axis: Variable Parameter eld of the
Control window once you select one of these parameters from the Parameter list.
As networks become larger the number of species also increases. By default, LiNA
plots the mean values of all species in the Plots window after startup, which can be
confusing. Species can be selected or deselected by pressing the corresponding species
button in the Variables eld of the Control window. For large systems it can be faster
to deselect all species rst and then select only those species that are of interest: Use the
right mouse button while holding the mouse pointer over the Variables eld name.
To generate the ultrasensitive response curves shown in Fig. 2.7, choose C_K from the
Parameter list in the X-Axis:Variable Parameter eld and change the value of C_Sp from
1 to 10 so that ST PT . Note that the condition max (K1 , K2 ) ST is already fullled.
Now, adjust the upper boundary of the Range to 2 and increase the number of Intervals
from 10 to 40. Observe that both, S (S ) and Sp (S ∗ ), exhibit a sharp transition near
C_K=1 (KT = 1). This suggests that the steady state uctuations of these two species
are large near C_K=1. Convince yourself that this is true: Specically, observe that the
Fano factors of S and Sp change by a factor of more than 20 near C_K=1 (Choose Fano
Factor from the pull-down list in the Y-Axis eld of the Control window). In contrast,
the coecient of variation, which measures ratio between the standard deviation and the
mean value, changes much less dramatically. (Choose Coecient of Variation from the
pull-down list in the Y-Axis eld of the Control window)
2.2.2. Correlation Analysis
As systems become larger it can be interesting to study correlations among the species.
As Figure 2.8 shows the strength of correlations changes as parameters are varied. There
are two ways to start the correlation analysis. First, you can select Correlation from the
pull-down list of the Y-Axis eld in the Control window. By default, this will only plot
the correlation coecient among the rst two species of the Variables eld. Alternatively,
you can press the Update Plot button in the Correlations eld of the Control window
which has the same eect.
To add new species pairs simply choose the corresponding species from the pull-down
lists in the Correlations eld and press Add. You may also select All species or all Selected
species at once and press Add. For example, the combination Species /Selected will add
all distinct pairs between Species and the selected species in the Variables eld of the
Control window. Similarly, the combination All /All would add all n(n − 1)/2 distinct
species pairs (where n is the total number of species in the system) to the list in the
Correlations eld. Species pairs in the Correlations eld can be highlighted by clicking
13
2. Tutorials
Figure 2.7.: Ultrasensitivity in the Goldbeter-Koshland model for K1 = K2 = 0.1µM ,
k1 = k2 = 1/s, ST (C_Sp) = 10µM and PT (C_P) = 1µM . Note that the
curves for SK and SpP overlap.
on them (press the Shift or Control key for multiple selection). Highlighted species are
removed via a context menu that opens upon a right-click with the mouse button within
the Correlations eld. This method is especially useful if you want to remove a small
number of species pairs. However, imagine you have all n(n − 1)/2 species pairs in the
list and you want to remove only those that contain a particular species, say X. In that
case it might be faster to choose the combination X /All from the pull-down list and
press the Remove button.
Despite the fact that the existence of a correlation between species A and B, in general,
does not imply a sense of interaction between them, the analysis of correlations can still
provide some hints for the functionality of a network. To this end, delete the S /K pair
from the Correlations list in the Control window and add the pairs S /Sp, K /Sp and
P /Sp and press Update Plot. As you can see the modied and the unmodied forms of
the substrate are always anti-correlated (Fig. 2.8) which is a direct consequence of the
approximate conservation relation for the substrate ST ≈ [S] + [S ∗ ], which holds under
subtrate excess. As the total substrate concentration is reduced this anti-correlation
becomes weaker. (Try reducing C_Sp!) The almost perfect correlation between S ∗ (Sp)
and K beyond the transition point C_K=1 means that essentially all of the modied
substrate is generated through the reaction S -K → S ∗ + K in the parameter region
C_K>1 and not through the dissociation of the S ∗ -P complex - in agreement with the
fact that the correlation coecient for S ∗ (Sp) and P is essentially zero for C_K>1.
14
2. Tutorials
Figure 2.8.: Correlation coecients for selected species pairs in the Goldbeter-Koshland
model (Eq. 2.5). Parameter values are the same as those used in Fig. 2.7.
2.3. Bistability in Covalent Modication Cycles
Here, we consider an extension of the Goldbeter-Koshland model which has been recently
analyzed in the context of bistability and hysteresis [8]. The elementary reactions read
k1+
k
S + KL S -KL →1 S ∗ + KL
k1−
S+P
k2+
(2.6)
k
S ∗ -P →2 S + P
k2−
k3+
K + L KL
k3−
k4+
P + L PL
k4−
where L represents an allosteric eector that reciprocally aects the activities of the
converter enzymes. Specically, binding of L is assumed to activate the kinase and to
inactivate the phosphatase.
It can be shown that the steady states of this system depend on the Michaelis-Menten
constants of the kinase (K1 = (k1 + k1− )/k1+ ) and the phosphatase (K2 = (k2 + k2− )/k2+ ),
on the dissociation constants of the enzyme-eector complexes KL (KD3 = k3− /k3+ )
and P L (KD4 = k4− /k4+ ), on the catalytic rate constants of the kinase (k1 ) and the
15
2. Tutorials
Abbildung 2.9.: Steady state response curve for the extended Goldbeter-Koshland model
in Eq. (2.6). In the region between the two saddle-node bifurcations
(SN) two stable steady states (solid lines) coexist with one unstable
steady state (dashed line). Parameter values: K1 = 1µM , K2 = 0.01µM ,
KD3 = 10µM , KD4 = 0.01µM , k1 = 10/s, k2 = 0.1/s, ST = 3µM ,
KT = PT = 1µM .
phosphatase (k2 ) as well as on the total concentrations of substrate (ST ) and converter
enzymes (KT , PT ). If certain conditions among these parameters are met this system
can exhibit bistability for a range of eector concentrations (Fig. 2.9).
2.3.1. Model Setup
Load the bistable_Goldbeter_Koshland_model.mat le from the Examples folder. Note
that, in the Parameter Substitutions eld, the `on' rate constants ki+ (i = 1, . . . , 4) have
been replaced by Michaelis-Menten (K1 , K2 ) and dissociation constants (KD3 , KD4 )
according to
ki+ =
ki + ki−
,
Ki
for i = 1, 2 and
ki+ =
ki−
,
KDi
for i = 3, 4.
After starting the calculations you will nd 4 new parameters in the Parameters list of the
Control window which correspond, respectively, to the total concentration of substrate
(C_Sp), converter enzymes (C_K and C_P) and allosteric eector (C_L). Convince
yourself that the steady states do not depend on the backward rate constants ki− (kim).
2.3.2. Strategies for Dealing with Multiple Steady States
LiNA cannot generate bifurcation diagrams, as in Fig. 2.9, directly which would require
continuation methods [6]. Instead, LiNA uses a simple strategy to nd the stable branches
of the steady state curve, either by using dierent starting points in parameter space or
by trying to numerically nd all solutions of the steady state equations.
16
2. Tutorials
First Strategy
To generate the stable branches of the steady state curve shown in Fig. 2.9, choose C_L
from the Parameter list in the X-Axis:Variable Parameter eld, change the values of k1
and C_Sp from 1 to 10 and 3, respectively, adjust the upper boundary of the Range to
0.5, increase the number of Intervals from 10 to 40 and deselect all species except for Sp
(S ∗ ). This should result in a plot as in Fig. 2.10.
Apparently, we have obtained the upper stable branch of the steady state curve in
Fig. 2.9 as well as the part of the lower branch that is to the left of the saddle-node
bifurcation (SN). The rst strategy to obtain the lower stable branch is to generate the
steady state curve in the opposite direction. This can be done using the Rescan buttons
in the Multiple Steady States eld of the Control window (A, Fig. 2.10). By repeatedly
using the left and right arrow you alternatively obtain either the lower or the upper part
of the steady state branch. (Try it!) If you wish to keep both branches press the copy
button right next to Solution branch: 1 in the Multiple Steady States eld. This will
create a copy of the current solution branch. Then, use again the Rescan buttons to
generate the complementary branch. Now you can access both branches independently
either through the Solution branch pull-down list in the Multiple Steady States eld or
through the pull-down list in the Plots window (B, Fig. 2.10). In the latter case, the
branch number is indicated in square brackets.
Second Strategy
If a system has more than two stable steady states it might be dicult, if not impossible,
to nd all stable branches with the rst strategy. In that case you may press the All
Steady States button in the Multiple Steady States eld of the Control window. This
function tries to nd all steady state curves numerically. It checks for local stability and
returns the stable branches which should then appear in the Solution branch pull-down
list. For the current example, All Steady States yields the same result as the strategy
using the Rescan buttons (Try it!). In general, it seems reasonable to combine both
strategies. The method using the Rescan buttons is fast and gives a rst indication
whether a system admits multiple stable steady states in a certain parameter range. All
Steady States can be used to test whether there are further stable branches which may
exist, for example, inside the region between two saddle-node bifurcations.
Having found the stable branches of the steady state set one may plot the Fano factor
or any of the other quantities from the pull-down list of the Y-axis eld along these
branches similar as in the case of a single steady state curve. Note, however, that the
predictions, based on the LNA, may become worse close to the saddle-node bifurcations
as the LNA fails at such bifurcation points.
17
2. Tutorials
Figure 2.10.: Stable branches of the steady state curve in Fig. 2.9. Parameter values are
the same as in the caption of Fig. 2.9.
18
3. Technical Remarks
To generate parameter scans for graphical visualization LiNA solves the steady state
equations that result from the linear noise approximation of the master equation (cf.
Eqs. 4.8 and 4.14)
d~x
dt
dσ
dt
= N · f~0 (~x, p~) = 0
(3.1)
= K (~x) · σ + σ · KT (~x) + D (~x) = 0.
(3.2)
For a given network topology (encoded by the stoichiometric matrix N) and a given set of
kinetic parameters (p~) Eq. (3.1) represents k , in general nonlinear, algebraic equations for
the unknown steady state concentrations of the k species ~xs = (xs1 , . . . , xsk ). In contrast,
Eq. (3.2) results in a set of linear algebraic equations for the k (k + 1) /2 independent
components of the variance-covariance matrix σ . Hence, for a given set of parameters
LiNA has to solve k + k (k + 1) /2 algebraic equations. From the values of ~xs and σ all
other quantities of interest can be derived (cf. Section 4.3).
Although the steady states are calculated numerically, at the beginning of the analysis
LiNA sets up Eqs. (3.1) and (3.2) symbolically in order to apply parameter substitutions
if desired. All symbolic calculations are performed with MuPAD which is available as
symbolic toolbox for MATLAB since release 2007b. In order to make use of the symbolic
results they are converted into character strings in a format which can be processed
by the MATLAB parser. For actual computations, parameters are substituted by their
values where necessary and the resulting strings are parsed into anonymous MATLAB
functions which then can be used as inputs for the numerical methods. In this manner
exact analytical results are used as far as possible and the numerical calculations can
still be performed eciently with the parsed functions.
3.1. Solving Eqs. (3.1) and (3.2)
The primary method for calculating steady states is the MATLAB root solver fsolve.
Only non-negative steady states that fulll the conservation relations (when present)
and are locally stable are accepted as valid. In case the root solver returns an invalid
steady state or fails to nd a solution the ODE system in Eq. (3.1) is numerically
integrated for a given amount of time and the resulting end point taken as a new starting
point for the root solver. This procedure is iterated up to three times with the integration
time increasing in each iteration. If an integration occurs a message is displayed at the
MATLAB console. In order to improve the eciency, a linear interpolation of already
known steady states is kept from which start points for the calculation of further steady
19
3. Technical Remarks
states are derived. To solve Eq. (3.2) the matrices K (~xs ) and D (~xs ) are evaluated at the
steady state solution ~xs found by solving Eq. (3.1). The equation is then transformed
into the standard form A · ~σ = ~b which can be readily solved using standard MATLAB
functions. The vector ~σ contains the independent components of the variance-covariance
matrix. For example, in the case of a reaction network with 2 species it would be given
by ~σ = (σ11 , σ22 , σ12 )T .
3.2.
Rescan
vs.
All Steady States
In case Eq. (3.1) admits multiple stable steady states one can use the functions Rescan
or All Steady States from the Control window to nd them. The Rescan function uses
the previous steady state as a starting point for calculating the next steady state along
the parameter curve (in scan direction). In contrast, the function All Steady States tries
to nd all stable steady states in given parameter range by calling the MuPAD function
numeric::solve. Although this is convenient in case of bifurcations it is also a quite timeconsuming procedure. Note that neither rescan nor calculation of all steady states will
introduce any new support points along the parameter curve but rather operates on those
points that are visible in the current plot window.
3.3. Data Management
The primary results of steady states computations, i.e. the mean values and variances, are
permanently kept by LiNA unless they are explicitly cleared. They are stored separately
for each combination of parameter set and variable parameter that has been selected at
any time. Covariances, which are needed for calculating correlations, are calculated for
all pairs that have been displayed in a correlation plot for a given parameter set/variabel
parameter combination at any time and will be kept from that point onwards.
3.4. Output at MATLAB Console and vanKampen.out
After starting the analysis by pressing the Start button in the Equation editor (cf.
Fig. 2.2) a message is displayed at the console indicating whether or not parameter
substitution (if intended) has been successful and whether conservation relations have
been detected. The console also displays the result of parameter scans each time a
parameter or a parameter range is changed. In case something goes wrong during the
computation an error message is displayed at the console which might help to trace
possible problems in the computation.
For the record, the results of the symbolic computations are written to the le vanKampen.out each time a new analysis is started. Specically, these are the stoichiometric
matrix N, the Jacobian K, the diusion matrix D as well as the matrix A and the
vector ~b which are used to solve Eq. (3.2). The le also displays the right-hand side of
the ODE system in Eq. (3.1) in symbolic form, lists the independent species together
with the reduced stoichiometric matrix NR and conservation relations (if present).
20
4. Theoretical Background
In this Chapter we provide the necessary theoretical background to appreciate the functionality implemented in LiNA. To make the presentation as self-contained as possible we
give a derivation of the linear noise approximation of the master equation for biochemical
reaction networks in Appendix A. We also describe how to deal with mass-conservation
relations. Specic examples are provided to illustrate the general method.
4.1. Reaction Networks and the Master Equation
We consider a volume V where r chemical reactions take place between k species. Such
a stochiometric reaction network can be written in the form
Wj
+
+
−
−
Xk ,
X1 + . . . + νkj
Xk → ν1j
ν1j
X1 + . . . + νkj
j = 1, . . . , r
(4.1)
±
where the index j numbers the reactions. From the stoichiometric coecients νij
∈
{0, 1, 2, . . .} one can construct the stoichiometric matrix N which has dimension k × r.
Its components, which are dened by
+
−
Nij := νij
− νij
,
i = 1, . . . , k,
j = 1, . . . , r ,
(4.2)
indicate how many molecules of species Xi are produced (Nij > 0) or consumed (Nij < 0)
in the j th reaction.
Let ni denote the number of molecules of species Xi . Then the state of the reaction
system is specied by the probability P (ni , t) to have ni molecules of species Xi at
time t in the system. In a Markovian description of the reaction network (Eq. 4.1) the
dynamics of P (ni , t) is determined by transition rates Wj (ni + Nij |ni ) which denote the
probability per unit time for a change of the number of molecules of species Xi by an
amount Nij as a result of reaction j . For given transition rates the temporal evolution of
P (ni , t) is determined by the chemical master equation [9]
r
i
dP (~n, t) X h
~ j P ~n − N
~ j , t − Wj ~n + N
~ j |~n P (~n, t)
=
Wj ~n|~n − N
dt
(4.3)
j=1
where ~n = (n1 , . . . , nk )T is a column vector whose components denote the molecule
~ j = (N1j , . . . , Nkj )T represent a set of vectors that are
numbers ni of species Xi and N
constructed from the j th column of the stoichiometric matrix as dened in Eq. (4.2).
Despite the fact that Eq. (4.3) is linear in P (~n, t) the transition rates Wj often exhibit a
nonlinear dependence on ~n so that exact solutions of this equation are rare. As a result,
one either relies on numerical simulations [2] or on approximation methods.
21
4. Theoretical Background
One such approximation method is the linear noise approximation (LNA) which is
based on van Kampen's system size expansion of the master equation [9]. To this end,
one assumes that the molecule number vector can be decomposed as
1
ni = Vm xi + Vm2 ξi ,
where
xi :=
(4.4)
i = 1, . . . , k
hni i
Vm
(4.5)
represents the average concentration of species Xi and
Vm ≡ V · NA
NA ≈ 6 · 1023 particles/mol
(4.6)
denotes the molar volume. Here, we have followed standard biochemical conventions
according to which species concentrations are measured in units of mol/liter.
The k quantities ξi in Eq. (4.4) represent the new stochastic variables describing the
deviations from the deterministic behavior. Hence, the idea of the decomposition in Eq.
(4.4) is to separate the average dynamics, described by xi , from the uctuations ξi which
are described by a new probability density Π (ξi , t).
4.2. The Linear Noise Approximation
The LNA is obtained by inserting the decomposition in Eq. (4.4) into the master equa−1
tion (Eq. 4.3) and expanding the resulting equation in powers of Vm 2 . The rationale
behind this procedure is that one expects that, in the limit of large system size V → ∞,
uctuations become negligible and one obtains an eective equation for the average concentration ~x. To perform
the expansion
one has to make an assumption about how the
~
transition rates Wj ~n + Nj |~n ≡ Wj (~n) in Eq. (4.3) depend on the system size. By
looking at specic examples (see Section 4.5) it turns out that most cases of practical
interest are covered by the Ansatz (cf. Ref. [9])
~n
Vm
~n
Wj (~n) = Vm f0,j
+ f1,j
Vm
∞
X
− 12
1−l
~
=
Vm fl,j ~x + Vm ξ
1
+
f2,j
Vm
~n
Vm
+ ...
(4.7)
l=0
which ensures that, in the limit V → ∞, the functions fl,j only depend on the average
concentration ~x and contributions from terms with l ≥ 2 become negligible for V → ∞.
Also, the assumption that the leading order term is proportional to V reects the fact
that the probability for a particular transition increases with the reaction volume if the
average concentration is kept constant. Later, we shall see that, in the LNA, the dynamics
of ~x and ξ~ in Eq. (4.4) is completely determined by the leading order term in Eq. (4.7).
22
4. Theoretical Background
For the average concentration ~x (t) the expansion of the master equation yields a system
of nonlinear ordinary dierential equations (ODEs) given by (cf. Section A.2)
r
dxi X
=
Nij f0,j (~x) ,
dt
i = 1, . . . , k
(4.8)
j=1
whereas the density of the uctuations around the
partial dierential equation
h
i

~ t
~ t
k
∂Π ξ,
∂ ξi0 Π ξ,
X

=
+
−Kii0 (~x)
∂t
∂ξi
0
i,i =1
average is described by the linear

2 Π ξ,
~ t
∂
1
 ,
Dii0 (~x)
2
∂ξi ∂ξi0
(4.9)
which is called Fokker-Planck equation . The drift matrix K and the diusion matrix D
in Eq. (4.9) are dened through
K (~x) :=
ii0
r
X
j=1
∂f0,j (~x)
Nij
∂xi0
and D (~x) :=
ii0
r
X
f0,j (~x) Nij Ni0 j .
(4.10)
j=1
The nomenclature `linear noise approximation', which is used to denote Eq. (4.9), results
from the fact that the coecients in front of the derivative terms in the Fokker-Planck
equation are, at most, linear in ξ~. From the denitions, given in Eq. (4.10), it is clear
that K represents the Jacobian matrix associated with the reaction system in Eq. (4.8).
In general, the coecient matrices K (~x) and D (~x) depend implicitely on time through
their dependence on the average concentration ~x(t). Hence, to perform the LNA one
rst has to solve the ODE system in Eq. (4.8). Of course, in most practical cases this
can only be done numerically. In a second step, one solves the Fokker-Planck equation
(4.9) where the mean-eld solution enters through the coecient matrices K (~x) and
D (~x). However, since the Fokker-Planck equation is linear with respect to ξ~ its solution
is always a multivariate Gaussian distribution given by


k
−1
X
(ξi − hξi (t)i) σii0 (t) (ξi0 − hξi0 (t)i)
1
~ t =q
 (4.11)
Π ξ,
exp −
2
k
0
i,i =1
(2π) det σ (t)
where σii−1
0 are the components of the inverse variance-covariance matrix
σii0 = hξi ξi0 i ,
i, i0 = 1, . . . , k
(4.12)
and det σ 6= 0 denotes its determinant. Using the Fokker-Planck equation in Eq. (4.9)
~ and σ are determined by the linear ODE systems
it is straightforward to show that hξi
(cf. Section A.3)
~
dhξi
dt
dσ
dt
~
= K · hξi
(4.13)
= K · σ + σ · KT + D
(4.14)
23
4. Theoretical Background
~ obey the same equation as small deviations from a steady
i.e. the average uctuations hξi
state of the mean-eld equations (Eq. 4.8). This means that the LNA is only applicable
near an asymptotically stable steady state where all eigenvalues of K have a negative
real part and the average uctuations decay to zero in time. If the mean-eld equations
(Eq. 4.8) exhibit multiple stable steady states one can perform the LNA near each of
the stable branches away from the bifurcation points.
The variance-covariance matrix, as dened in Eq. (4.12), has the same units as the
average concentration ~x (cf. Eq. 4.4), i.e. mol/liter in our case. However, if one
is interested in particle number uctuations rather than concentration uctuations one
has to transform the Gaussian distribution in Eq. (4.11) back to original variables.
Specically, from Eq. (4.4) the following relations are apparent
ξi =
ni − Vm xi
1
Vm2
1
ξi − hξi i =
ni − Vm xi − Vm2 hξi i
1
2
≡
ni − hni i
Vm
1
Vm2
−k
dξ~ = Vm 2 d~n
so that the Gaussian distribution in Eq. (4.11) can be rewritten as


k
−1
X
(ni − hni (t)i) Cii0 (t) (ni0 − hni0 (t)i)
1

exp −
P (~n, t) = q
2
k
i,i0 =1
(2π) det C (t)
where
Cii0 = Vm σii0
(4.15)
denotes the variance-covariance matrix with respect to particle numbers. The dynamics
of C follows from that of σ by multiplying Eq. (4.14) by Vm which results in
dC
= K · C + C · KT + V m D .
dt
4.3. Calculation of Stochastic Quantities
From the variance-covariance matrix σ (Eq. 4.14) and the mean-eld solution ~x (4.8) one
can readily calculate several stochastic quantities of interest such as the Fano factor Fi
and the coecient of variation CVi of the species Xi as well as the correlation coecient
rii0 between species Xi and Xi0 . Using Eqs. (4.4) and (4.15) these quantities can be
expressed as
Cii
σii
=
, i = 1, . . . , k
(4.16)
Fi =
hni i
xi
√
√
σii
Cii
CVi =
=√
, i = 1, . . . , k
(4.17)
hni i
V m xi
σii0
rii0 = √ √
, i, i0 = 1, . . . , k .
(4.18)
σii σi0 i0
24
4. Theoretical Background
This shows that, within the LNA, the Fano factor and the correlation coecients are
−1
independent of the reaction volume V . In particular, since ξi ∼ Vm 2 the components of
the variance-covariance matrix σii0 = hξi ξi0 i have the same unit as the mean concentrations xi . Also, from Eq. (4.17) we
p see that the standard deviation with respect to the
mean concentration is given by σii /Vm which has the same unit as xi . In contrast,
the coecient of variation depends inversely on the molar volume and, hence, the CV
vanishes as the reaction volume V increases.
4.4. Dealing with Mass Conservation
Species that are not actively synthesized or degraded obey mass conservation relations
which are indicated by linear dependencies between some rows of the stoichiometric
matrix N. This may happen, for example, if a transcription factor binds to specic
target promoters on the DNA in which case the promoters can exist in two states: free
or occupied by a transcription factor, but the total amount of promoters is constant in
time, at least over the time scale of the cell cycle. Another important example, where
mass conservation must be taken into account, is for the modeling of enzyme-catalyzed
reactions under in vitro conditions. Here, substrates, enzymes and cofactors are typically
supplied in xed amounts so that their total concentrations remain constant in time.
In general, there are as many linearly dependent rows of N as there are conserved
species in the system. Mathematically, mass conservation relations are associated with
the left null space of N which is dened by
S · N = 0.
(4.19)
The dimension of S is (k − k 0 ) × k where k 0 is the number of linearly independent rows of
N, i.e. S has a many rows as there are conserved species in the system, and the number
of columns equals the number of species. In terms of S one can dene the vector of
conserved species ~s through
~s = S · ~x .
(4.20)
Using the mean-eld equations in Eq. (4.8) it is straightforward to see that ~s is conserved
in time, i.e.
d~s
d~x
=S·
= S · N · f~0 (~x) = 0 ,
(4.21)
dt
dt
which means that
~s (t) = S · ~x (t) ≡ ~s0 ∀t
(4.22)
where ~s0 is a constant vector whose components denote the total (molar) concentrations
of the conserved species. Writing Eq. (4.22) in terms of particle numbers, rather than
concentrations, yields
S · ~n (t) = ~n0
(4.23)
where ~n0 = Vm~s0 now denotes total particle numbers of the conserved species.
25
4. Theoretical Background
Remark
For ~s0 (or ~n0 ) to be interpretable as a vector of total concentrations (or particle numbers)
it is necessary that all elements of S are positive. However, solving Eq. (4.19) generally
yields a matrix S which contains both positive and negative entries. Hence, before
dening ~s through Eq. (4.20), one rst has to nd a positive basis of the left null space
of N, e.g. through appropriate linear combinations of the rows of S.
The conservation relations in Eq. (4.23) are k − k 0 linear equations for the k particle
numbers of species Xi (i = 1, . . . , k ), which can be used to partition the Xi into k 0
independent species XR,i (i = 1, . . . , k 0 ) and k − k 0 dependent species Yα (α = 1, . . . , k −
k 0 ). Let's denote by nR,i (xR,i ) and mα (yα ) the particle numbers (concentrations)
associated with species XR,i and Yα , respectively. Then, the repartioned quantities can
be obtained from the original quantities according to
~nR
~xR
0
0
~n ≡
= P · ~n and ~x ≡
= P · ~x
(4.24)
m
~
~y
where P is an appropriate permutation matrix. Using these relations in conjunction
with Eq. (4.4) one can also decompose the uctuations into those associated with the
independent species (ξ~R ) and those associated with the dependent species (~η ) as
ξ~R
0
~
ξ =
= P · ξ~
(4.25)
~η
so that particle number vectors of the dependent and the independent species can be
written in the form
1
~nR = Vm ~xR + Vm2 ξ~R
(4.26)
1
2
m
~ = Vm ~y + Vm ~η .
In the next step, we solve Eqs. (4.22) and (4.23) for ~y and m
~ , respectively, which allows
expressing the dependent quantities as linear combinations of the independent quantities
(~xR and ~nR ) through the so-called link matrix L [7] dened by
~y = ~s0 + L · ~xR
(4.27)
m
~ = n0 + L · ~nR .
(4.28)
Inserting the decompositions from Eq. (4.26) into Eq. (4.28) and comparing with Eq.
(4.27) shows that dependent and independent uctuations are related as
~η = L · ξ~R .
(4.29)
Finally, one obtains the dynamics of the reduced system in the form
d~xR
= NR · f~0 (~xR ; ~s0 )
dt
26
(4.30)
4. Theoretical Background
where the reduced stoichiometric matrix NR is obtained by taking the rst k 0 rows of the
matrix P · N and the reaction rate vector f~0 is evaluated at the conservation relations
using Eqs. (4.24) and (4.27). Similarly, we dene the reduced drift matrix KR and the
reduced diusion matrix DR by
KR,ii0 (~xR ; ~s0 ) :=
DR,ii0 (~xR ; ~s0 ) :=
r
X
j=1
r
X
NR,ij
∂f0,j (~xR ; ~s0 )
∂xR,i0
and
(4.31)
f0,j (~xR ; ~s0 ) NR,ij NR,i0 j
j=1
where the indices range between i, i0 = 1, . . . , k 0 . Using KR and DR the dynamics of the
variance-covariance matrix of the reduced system is determined by
dσR
= KR · σR + σR · KTR + DR
(4.32)
dt
where the components of σR are given by σR,ii0 = ξR,i ξR,i0 for i, i0 = 1, . . . , k 0 . Hence,
one can apply the linear noise approximation to the reduced system, dened by Eq.
(4.30), in a similar way as described in Section 4.2.
The components of the variance-covariance matrix associated with the conserved species
can be calculated from the reduced variance-covariance matrix σR and Eq. (4.29). Specifically, one obtains
* k0
+
k0
X
X
0
σ(k
= hηα ηβ i =
Lαi ξR,i
Lβi0 ξR,i0
0 +α)(k 0 +β)
i=1
=
k0
X
Lαi Lβi0 σR,ii0 ,
i0 =1
α, β = 1, . . . , k − k 0
(4.33)
i,i0 =1
where σ 0 denotes the variance-covariance matrix with respect to the transformed uctuation vector ξ~0 dened in Eq. (4.25). Similarly, the `mixed' components are obtained
from
!
+
* k0
X
σ(k0 +α)(i) = hηα ξR,i i =
Lαi0 ξR,i0 ξR,i
i0 =1
k0
=
X
Lαi0 σR,i0 i ,
i = 1, . . . , k 0 ,
α = 1, . . . , k − k 0 .
(4.34)
i0 =1
4.5. Examples
In the next two subsections we demonstrate how to apply the linear noise approximation
to specic reaction systems. In both cases it is possible to derive explicit expressions for
the steady state uctuations. In addition, the example in subsection 4.5.2 shows how to
deal with mass-conservation relations.
27
4. Theoretical Background
4.5.1. Noise Reduction Through Receptor Dimerization
As a rst example, we consider the reaction scheme
ks
(4.35)
∅ R (X1 )
kd1
k+
(4.36)
R (X1 ) + R (X1 ) R2 (X2 )
k−
kd2
(4.37)
R2 (X2 ) → ∅
which describes synthesis and degradation of a receptor R (Eq. 4.35) and its dimerization
(Eq. 4.36). In addition, we assume that the dimerized receptor (R2 ) can also be degraded
with a specic rate that may dier from that for the monomer (4.37). First, we note
that in the absence of dimerization independent synthesis and degradation of receptor
molecules, as described by Eq. (4.35), leads to Poissonian steady state uctuations where
2 , so that the Fano factor
the average concentration hRi = ks /kd1 is equal to its variance σR
2 / hRi becomes 1. Moreover, Hayot and Jayaprakash observed [5] that, within
F1 = σR
2 / hR i = 1. Here, we
the LNA, this holds also true for receptor dimers, i.e. F2 = σR
2
2
show that a non-vanishing dimer degradation rate (kd2 6= 0) leads to sub-Poissonian
uctuations for both, monomers and dimers. In addition, we nd that the steady state
uctuations of dimerized receptors can become lower than those for monomers (F2 < F1 )
under certain conditions.
The stoichiometric matrix for the system in Eqs. (4.35) - (4.37) is given by
R
1 −1 −2 2
0
N=
.
(4.38)
R2
0 0
1 −1 −1
Hence, there are ve stoichiometric vectors
−2
−1
1
~
~
~
,
, N3 =
, N2 =
N1 =
1
0
0
~4 =
N
2
−1
,
~5 =
N
0
−1
,
but no conservation relations.
Let's denote the number of receptor monomers and dimers by n1 and n2 , respectively.
Then the ve transition rates are given by
~1
~ 2 = kd1 n1
W1 ~n|~n − N
= k̄s , W2 ~n|~n − N
~3
~ 4 = k − n2
W3 ~n|~n − N
= k̄ + n1 (n1 − 1) , W4 ~n|~n − N
~5
W5 ~n|~n − N
= kd2 n2
where ~n = (n1 , n2 )T . The units of k̄s and k̄ + are
[k̄s ] =
particles
s
and [k̄ + ] =
28
1
particles · s
4. Theoretical Background
whereas the unit of the remaining three constants is
1
[kd1 ] = [k − ] = [kd2 ] = .
s
Next, we expand the transition rates in the form (cf. Eq. 4.7)
Wj (~n) = Vm f0,j (~x) + f1,j (~x) + . . . ,
~x =
~n
.
Vm
For W2 , W4 and W5 this yields
W2 = Vm kd1 x1 ,
W4 = Vm k − x2
and W5 = Vm kd2 x2
f0,4 (~x) = k − x2
and f0,5 (~x) = kd2 x2
which shows that
f0,2 (~x) = kd1 x1 ,
(4.39)
and fl,j (~x) ≡ 0 for l ≥ 1 and j = 2, 4, 5. Hence, rst-order rate constants are unaected
when going from a mesoscopic to a macroscopic description.
However, expansion of the remaining transition rates
W1 = Vm ks
and W3 = k̄ + Vm x1 (Vm x1 − 1) ≡ Vm k + x21 − k + x1
(4.40)
shows that zero-order rate constants become inversely proportional to the reaction volume
(ks = k̄s /Vm ) whereas second-order rate constants become proportional to the reaction
volume (k + = Vm k̄ + ) so that ks and k + , as dened in Eqs. (4.35) and (4.36), now have
typical `macroscopic' units
[ks ] =
mol
M
≡
liter · s
s
and
liter
1
k+ =
≡
.
mol · s
M ·s
From Eq. (4.40) we infer that
f0,1 (~x) = ks
and f0,3 (~x) = k + x21
(4.41)
with fl,1 (~x) ≡ 0 for l ≥ 1 and fl,3 (~x) ≡ 0 for l ≥ 2. We also see that, as a result of the
dimerization reaction, there appears an O (1)-term (f1,3 (~x) = −k̄ + x1 ) in the expansion
for W3 which is, however, unimportant for the linear noise approximation.
(0)
The functions fj , dened in Eqs. (4.39) and (4.41), are used to construct the reaction
rate vector
T
(4.42)
f~0 = ks , kd1 x1 , k + x21 , k − x2 , kd2 x2 .
Then, the dynamics of the average concentration ~x = (x1 , x2 )T is determined by the
ODE system (cf. Eq. 4.8)
d~x
= N · f~0 (~x)
dt
29
4. Theoretical Background
or, in components,
dx1
dt
dx2
dt
= ks − kd1 x1 − 2k + x21 + 2k − x2
(4.43)
= k + x21 − k − + kd2 x2 .
For the matrices K and D, dened in Eq. (4.10), we obtain
−kd1 − 4k + x1
2k −
K=
2k + x1
− (k − + kd2 )
(4.44)
and
D = N · diag f~0 · NT
ks + kd1 x1 + 4k − x2 + 4k + x21
−2k − x2 − 2k + x21
=
(4.45)
−2k − x2 − 2k + x21
k + x21 + k − x2 + kd2 x2
where diag f~0 is a diagonal matrix which has the components f0j on its diagonal.
The solution of the steady state equations
ks − kd1 x1 − 2k + x21 + 2k − x2 = 0
k + x21 − k − + kd2 x2 = 0 ,
(4.46)
which result from Eq. (4.43) by setting dx1 /dt = 0 = dx2 /dt, is given by
xs2 =
xs1 =
where
Rs =
ks
,
kd1
α=
1
α (Rs − xs1 )
2 s
γ
2
1+
kd1
,
kd2
and
!
(4.47)
4Rs
−1
γ
β=
kd2
,
k−
(4.48)
KD =
k−
k+
(4.49)
and
α
.
2
Note that in the limit 4Rs γ expansion of the square root in Eq. (4.48) yields to
leading order
Rs2
xs1 ≈ Rs and xs2 ≈
.
(4.50)
KD (1 + β)
γ ≡ KD (1 + β)
Hence, by increasing the dissociation constant KD (weak association of monomers) or
increasing β (fast dimer degradation) one recovers the case of simple receptor synthesis
and degradation described by Eq. (4.35) where xs1 ≈ Rs and xs2 ≈ 0. However, in general
one expects that dimer dissociation is much faster than dimer degradation so that β 1.
30
4. Theoretical Background
To nd the three independent components of the variance-covariance matrix
σ11 σ12
σ=
σ12 σ22
we have to solve the system of linear equations dened by
K (~xs ) · σ + σ · K (~xs )T + D (~xs ) = 0
(4.51)
where the matrices K and D, dened in Eqs. (4.44) and (4.45), are now evaluated at
the steady state (cf. Eqs. 4.47 and 4.48). The solution of Eq. (4.51) is straightforward
although it is typically not easy to interpret. However, the resulting expressions can be
substantially simplied if one applies algebraic manipulations while repeatedly making
use of the steady state relations in Eq. (4.46). The result is
(1 + β (1 + α)) xs1 /KD + 4βxs2 /KD
s
s
(4.52)
σ11 = x1 1 −
(1 + β (1 + α) + 4xs1 /KD ) (α (1 + β) + 4xs1 /KD )
!
(2xs1 /KD )2
s
σ22
= xs2 1 −
(4.53)
(1 + β (1 + α) + 4xs1 /KD ) (α (1 + β) + 4xs1 /KD )
s
σ12
= −2
(1 + β) xs2 xs1 /KD
.
(1 + β (1 + α) + 4xs1 /KD ) (α (1 + β) + 4xs1 /KD )
(4.54)
From Eqs. (4.52) and (4.53) we see that receptor dimerization leads to sub-Poissonian
s /xs < 1) and receptor dimers (F =
uctuations for both receptor monomers (F1 = σ11
2
1
s /xs < 1). Similarly, Eq. (4.54) tells us that uctuations in receptor monomers are
σ22
2
s < 0. This is, of
always anti-correlated with uctuations in receptor dimers since σ12
course, not surprising since an increase of the dimer concentration must be accompanied
by a decrease in the concentration of monomers. However, since a decrease in dimers
does not necessarily result from an increase in monomers, but could also result from a
degradation of dimers, the strength of the anti-correlation will vary with parameters.
To estimate how much the intrinsic noise can be reduced by receptor dimerization we
consider the special case where α = 1 (monomers and dimers are degraded at the same
rate) and β 1. In that case the Fano factors can be approximated by
F1 ≈ 1 −
1
4xs1 /KD
4 (1 + 4xs1 /KD )2
(4.55)
F2 ≈ 1 −
1 (4xs1 /KD )2
4 (1 + 4xs1 /KD )2
(4.56)
which implies that
KD
.
4
In particular, one can show that, in the limit xs1 KD /4, F1 → 1 whereas F2 → 3/4.
Hence, dimerization can reduce the
√steady state uctuations of dimerized receptors if Rs
is suciently large. Indeed, xs1 ∼ γRs for Rs γ/4 (cf. Eq. 4.48).
F2 < F1 ,
for xs1 >
31
4. Theoretical Background
Finally, we note that the case kd2 = 0, studied in Ref. [5], is recovered by letting
in Eqs. (4.52) - (4.54) α → ∞ and β → 0 while keeping the product α · β = kd1 /k −
constant (cf. Eq. 4.49) which leads to xs1 → Rs and xs2 → Rs2 /KD (cf. Eq. 4.50) so
s → xs , σ s → xs ) and
that uctuations in monomers and dimers become Poissonian (σ11
1
22
2
s → 0).
correlations between monomers and dimers vanish (σ12
4.5.2. Receptor-Ligand Binding with Mass Conservation
As a second example, we consider the simple reaction scheme
k1+
L (X1 ) + R (X2 ) R-L (X3 )
k1−
which describes the binding of a ligand (L) to a receptor (R) under
The transition rates are given by
W1 = k̄1+ n1 n2
and
W2 = k1− n3
(4.57)
in vitro conditions.
(4.58)
−
where k̄1+ and k1 have the units 1/ (particle · s) and 1/s, respectively. The stoichiometric
matrix of the system in Eq. (4.57) reads


R
−1 1
N = L  −1 1  .
R -L
1 −1
In this example, we neglect reactions for synthesis and degradation of the ligand and
receptor molecules so that their total numbers remain constant. Consequently, two of
the three rows of N are linearly dependent. A positive basis of the left null space of N
is given by
1 0 1
S=
0 1 1
and the corresponding conservation relations read
x1 (t) + x3 (t)
LT
= ~s0 ≡
S · ~x (t) =
.
x2 (t) + x3 (t)
RT
(4.59)
We choose X3 as the independent, and X1 and X2 as the dependent species, i.e. we set
(cf. Eqs. 4.26)



 

xR
x1
x3
 y1  = P ·  x2  =  x1 
(4.60)
y2
x3
x2
where P denotes the permutation matrix (cf. Eq. 4.24)


0 0 1
P =  1 0 0 .
0 1 0
32
4. Theoretical Background
The uctuations are rearranged as

 



ξ3
ξ1
ξR
 η 1  = P ·  ξ2  =  ξ1  .
ξ2
ξ3
η2
(4.61)
Using Eq. (4.59) we can express the dependent species concentrations as a function of
the independent species concentration through
y1 = LT − xR
(4.62)
y2 = RT − xR
or
~y = ~s0 + L · ~xR
where the (2 × 1) link matrix L, as dened in Eq. (4.27), is given by
−1
L=
.
−1
(4.63)
Hence, uctuations of the dependent species can be obtained from that of the independent
species through (cf. Eq. 4.29)
η1 = −ξR
(4.64)
η2 = −ξR .
The dynamics of the reduced system is then determined by
dxR
= NR · f~0 (xR , ~s0 ) = k1+ (LT − xR ) (RT − xR ) − k1− xR
dt
(4.65)
where the reduced stoichiometric matrix is obtained from the rst row of P · N as
NR = 1 −1 .
The reaction rate vector is obtained by rst expanding the transition rates, dened in
Eq. (4.58), as
W1 = Vm k1+ x1 x2 ≡ Vm k1+ y1 y2
W2 = Vm k1− x3 ≡ Vm k1− xR
and then using the conservation relations in Eqs. (4.62) to replace y1 and y2 by xR and
~s0 = (LT , RT )T leading to
+
k1 (LT − xR ) (RT − xR )
~
f0 (xR , ~s0 ) =
.
k1− xR
In Eq. (4.65) the second-order rate constant k1+ has the unit liter/ (mol · s) ≡ 1/ (M · s).
33
4. Theoretical Background
Since the reduced system is 1-dimensional there is only 1 independent component of
the variance-covariance matrix which is determined by
σR = −
DR
2KR
where σR = hξR ξR i = hξ3 ξ3 i ≡ σ33 (cf. Eq. 4.61). The reduced Jacobian and diusion
matrices (KR and DR ) are given by
KR = −k1+ (LT + RT − 2xR ) − k1−
DR = k1+ (LT − xR ) (RT − xR ) + k1− xR .
Hence, under steady state conditions we have
!
k1+ (LT − xR ) (RT − xR ) = k1− xR
and the variance of the receptor-ligand complex can be written as
σR =
=
1 k1+ (LT − xR ) (RT − xR ) + k1− xR
2
k1+ (LT + RT − 2xR ) + k1−
xR
LT + RT − 2xR + KD
(4.66)
where xR is determined by the quadratic equation
x2R − xR (LT + RT + KD ) + LT RT = 0
(4.67)
and KD = k1− /k1+ denotes the dissociation constant of the receptor-ligand complex. The
biologically feasable solution of Eq. (4.67) is given by
!
s
LT + R T + K D
LT RT
xR =
1− 1−4
.
(4.68)
2
(LT + RT + KD )2
The remaining components of the variance-covariance matrix can be calculated using
the expressions in Eqs. (4.61) and (4.64) which follow from the general expressions in
Eqs. (4.33) and (4.34). For example, the variances of ligand and free receptors are
σ11 = hξ1 ξ1 i = hη1 η1 i = h(−ξR ) (−ξR )i = σR
σ22 = hξ2 ξ2 i = hη2 η2 i = h(−ξR ) (−ξR )i = σR .
Hence, the variances of all three species are the same (σ11 = σ22 = σ33 ≡ σR ). Similary,
the calculation of the covariances yields
σ12 = hξ1 ξ2 i = hη1 η2 i = h(−ξR ) (−ξR )i = σR
σ13 = hξ1 ξ3 i = hη1 ξR i = h(−ξR ) ξR i = −σR
σ23 = hξ2 ξ3 i = hη2 ξR i = h(−ξR ) ξR i = −σR .
34
4. Theoretical Background
This shows that the correlation coecients, as dened in Eq. (4.18), are given by
r13 = r23 = −1 and r12 = 1
which expresses the obvious fact that the disappearance of a ligand or receptor molecule
is always accompanied by the appearance of a receptor-ligand complex.
To obtain explicit expressions for other stochastic quantities such as the Fano factor
one may use the expressions for xR and σR in Eqs. (4.68) and (4.66). Alternatively,
one can try to obtain simpler expressions for xR from suitable approximate solutions of
Eq. (4.67). Specically, if KD RT or KD LT the solution of Eq. (4.67) can be
approximated by the discontinuous function

 LT 1 − KD
LT < RT
RT −LT xR ≈
(4.69)
K
D
 RT 1 −
RT < LT .
L −R
T
T
In the opposite case, when KD RT or KD LT one obtains
xR ≈
LT RT
.
LT + RT + KD
(4.70)
Hence, in the rst case (Eq. 4.69) the Fano factors are given to leading order by
(
1
LT < RT
σR
F1 =
≈
(4.71)
RT KD
LT − x R
R
<
L
2
T
T
(LT −RT )
( L K
T D
LT < RT
σR
(RT −LT )2
≈
F2 =
RT − xR
RT < LT
1
(
KD
LT < RT
σR
RT −LT
≈
F3 =
KD
xR
RT < LT ,
LT −RT
whereas in the second case (Eq. 4.70) they are given to leading order by
F1 ≈
F2 ≈
F3 ≈
KD RT
(LT + KD ) (LT + RT + KD )
KD LT
(RT + KD ) (LT + RT + KD )
KD
.
LT + RT + KD
(4.72)
From the expressions in Eqs. (4.71) we see that if receptor-ligand binding is tight, so
that KD min (RT , LT ), F1 exhibits a switch-like transition from F1 ≈ 1 for LT < RT
(Poissonian noise) to F1 ≈ 0 for LT > RT (noise suppression) and vice versa for F2 .
Problem
Check the validity of the approximate expressions in Eqs. (4.69) - (4.72) with LiNA!
35
A. Derivation of the Linear Noise
Approximation
In the following, we give a derivation of the dynamic equations
for the mean-eld ~x (Eq.
~
4.8) and the probability density of the uctuations Π ξ, t (Eq. 4.9), which are derived
from the master equation (Eq. 4.3) using the decomposion of the particle number vector
in Eq. (4.4). The derivation is a multivariate extension of that given in Ref. [9] for the
special case of biochemical reaction networks where the jump sizes in the transition rates
are determined by the entries of the stoichiometric matrix (cf. Eq. 4.3). An alternative
derivation of the multivariate linear noise approximation, based on the step operator
formalism, can be found in Ref. [1]. A derivation that goes beyond the linear noise
approximation has been given by Grima in Ref. [4].
A.1. Transformation of the Master Equation
~ t are related through
The probability densities P (~n, t) and Π ξ,
1
−1
2~
~ t
P (~n, t) = P Vm ~x + Vm ξ, t = Vm 2 Π ξ,
(A.1)
which follows from the invariance of the probability measure
1
!
~ t dξ~
P (~n, t) d~n = P (~n, t) Vm2 dξ~ = Π ξ,
and Eq. (4.4). The probability density at the shifted argument can be obtained from
1
− 12
2
~
~j, t
~j , t
P ~n − N
= P Vm ~x + Vm ξ − Vm N
− 12
− 12
~
~
= Vm Π ξ − Vm Nj , t .
(A.2)
36
A. Derivation of the Linear Noise Approximation
Under the transformation in Eq. (4.4) the time-derivative in Eq. (4.3) becomes


~ t
∂Π
ξ,
1
dP (~n, t)
−2


= Vm
dt
∂t
~
n





~
~ t
k
∂Π
ξ,
t
∂Π
ξ,
X
dξ
− 21
i

 +


= Vm 
∂t
∂ξi
dt ~n
i=1
t
ξ~





~
~
k
∂Π
ξ,
t
∂Π
ξ,
t
1
1 X
−
 − Vm2

 dxi 
= Vm 2 
∂t
∂ξi
dt
i=1
ξ~
(A.3)
t
where the last step follows from Eq. (4.4) if ~n is kept constant.
Next, we have to transform the transition rates in Eq. (4.3). To this end we recall the
expansion from Eq. (4.7)
Wj (~n) =
∞
X
Vm1−l fl,j
− 12
~
~x + Vm ξ
l=0
which can be used to obtain the transition rates at the shifted argument as
∞
X
− 12
− 12
1−l
~
~
~
Vm fl,j ~x + Vm
ξ − V m Nj
.
Wj ~n − Nj =
(A.4)
l=0
Inserting the expressions in Eqs. (A.1) - (A.4) into the master equation (Eq. 4.3) yields
1
(after multiplication by Vm2 )
~ t
∂Π ξ,
∞
X
l=0
Vm1−l
r X
−1
fl,j ~x + Vm 2
~ t
k ∂Π ξ,
X
dxi
− Vm
∂t
∂ξi
dt
i=1
− 12
− 21
~
~
~
~
Π ξ − Vm Nj , t
ξ − V m Nj
1
2
(A.5)
(A.6)
j=1
−1
~ t .
− fl,j ~x + Vm 2 ξ~ Π ξ,
The idea of
=
(A.7)
van Kampen's system size expansion is to expand the right-hand side of this
−1
master equation in powers of Vm 2 . Note that the left-hand side of the master equation
−1
(Eq. A.5) has already
the form
of an expansion in powers of Vm 2 ≡ ε which consists of
terms of O ε0 and O ε−1 . In order to match these terms by corresponding terms on
the right-hand side of the master equation (Eqs. A.6 and A.7) we expand the functions
− 12
− 21
− 12
~
~
~
~
fl,j ~x + Vm
ξ − V m Nj
Π ξ − Vm Nj , t
37
A. Derivation of the Linear Noise Approximation
1
1
−
~ j in the argument ξ~ − Vm− 2 N
~ j which leads to
with respect to Vm 2 N
− 12
− 12
− 12
~
~
~
~
fl,j ~x + Vm
Π ξ − Vm Nj , t
ξ − Vm Nj
−1
~ t
= fl,j ~x + Vm 2 ξ~ Π ξ,
− 21
~
~ t
x + Vm ξ Π ξ,
k ∂ fl,j ~
− 12 X
− Vm
Nij
∂ξi
i=1
− 21
2
~
~ t
fl,j ~x + Vm ξ Π ξ,
k ∂
1 −1 X
− 32
Nij Ni0 j + O Vm
+
V
2 m 0
∂ξi ∂ξi0
(A.8)
(A.9)
(A.10)
i,i =1
Note that the lowest order term in Eq. (A.8) cancels with the term in Eq. (A.7)
of
− 12
the master equation. The expressions in Eqs. (A.9) and (A.10) are of O Vm
and
1 −l
and O Vm−l to the
O Vm−1 , respectively, so that they contribute terms of O Vm2
master equation. However, since in Eq. (A.5) the lowest order term is of O (1) the terms
coming from Eqs. (A.9) and (A.10) can only be matched if l = 0.
−1
The arguments of the functions fl,j in Eqs. (A.9) and (A.10) still depend on Vm 2
−1
which, after expansion with respect to Vm 2 ξ~
k
x)
− 12
− 1 X ∂fl,j (~
fl,j ~x + Vm ξ~ = fl,j (~x) + Vm 2
ξi + O Vm−1 ,
∂xi
(A.11)
i=1
can potentially contribute to the matching with Eq. (A.5). However, for the second
order
−3
term in Eq. (A.10) the higher order terms in Eq. (A.11) would yield terms of O Vm 2
and higher which, thus, cannot be matched with the terms in Eq. (A.5). Hence, we may
write Eq. (A.10) in the form
− 21
2
~
~ t
fl,j ~x + Vm ξ Π ξ,
k ∂
1 −1 X
V
Nij Ni0 j
2 m 0
∂ξi ∂ξi0
i,i =1
2 Π ξ,
~ t
k
∂
1 −1 X
− 32
=
V
fl,j (~x) Nij Ni0 j
+ O Vm
.
(A.12)
2 m 0
∂ξi ∂ξi0
i,i =1
In contrast, for the rst order term in Eq. (A.9), the rst two terms in Eq. (A.11) give a
38
A. Derivation of the Linear Noise Approximation
contribution so that we write this term in the form
− 12
~ t
x + Vm ξ~ Π ξ,
k ∂ fl,j ~
− 12 X
Vm
Nij
∂ξi
i=1
~
k
1 X ∂Π ξ, t
−
Nij fl,j (~x)
= Vm 2
∂ξi
i=1
h
i
~ t
k
0
∂
ξ
Π
ξ,
X
i
∂fl,j (~x)
− 32
−1
+ Vm
Nij + O Vm
.
∂xi0
∂ξi
0
(A.13)
i,i =1
Finally, using the expressions in Eqs. (A.12) and (A.13) in Eqs. (A.10) and (A.9) and,
the resulting expression in Eq. (A.6) gives
~ t
~
~
k
r
k
∂Π ξ,
1 X ∂Π ξ, t dx
1 X X ∂Π ξ, t
i
2
2
− Vm
= −Vm
Nij f0,j (~x)
(A.14)
∂t
∂ξi
dt
∂ξi
i=1
j=1 i=1
h
i
~ t
r X
k
0
∂
ξ
Π
ξ,
X
i
∂f0,j (~x)
−
Nij
∂xi0
∂ξi
j=1 i,i0 =1
2 Π ξ,
~ t
r
k
∂
X
X
1
− 12
+
+ O Vm
f0,j (~x)
Nij Ni0 j
2
∂ξi ∂ξi0
0
j=1
i,i =1
where we have retained only the l = 0 term.
A.2. Eective Equations for ~x (t) and
~ t
Π ξ,
Comparing rst the terms of highest order in Eq. (A.14) gives


1
~ t
k ∂Π ξ,
r
X
X
 dxi −
O Vm2 :
Nij f0,j (~x) = 0 .
∂ξi
dt
i=1
j=1
This implies that ~x has to fulll the mean-eld equations (cf. Eq. 4.8)
r
dxi X
=
Nij f0,j (~x) ,
dt
i = 1, . . . , k
j=1
provided that
~ t
∂Π ξ,
∂ξi
6= 0,
39
i = 1, . . . , k
(A.15)
A. Derivation of the Linear Noise Approximation
which means that Π must not be constant with respect to ξ~ or, in other words, the
mean-eld equation (Eq. A.15) is only valid if there are uctuations in the system.
Comparing the terms at next order yields a Fokker-Planck equation for the density of
the uctuations given by
h
i
~ t
~ t
r X
k
0
∂Π ξ,
∂
ξ
Π
ξ,
X
i
∂f0,j (~x)
O (1) :
(A.16)
= −
Nij
∂t
∂xi0
∂ξi
j=1 i,i0 =1
2 Π ξ,
~ t
r
k
∂
X
X
1
+
.
f0,j (~x)
Nij Ni0 j
2
∂ξi ∂ξi0
0
j=1
i,i =1
Introducing the drift matrix K and the diusion matrix D by
Kii0 (~x) :=
r
X
j=1
∂f0,j (~x)
Nij
∂xi0
and
Dii0 (~x) :=
r
X
f0,j (~x) Nij Ni0 j
(A.17)
j=1
the Fokker-Planck equation can be written in compact notation
h
i


2 Π ξ,
~
~ t
~ t
k
0
∂
ξ
Π
∂
ξ,
t
∂Π ξ,
X
i
1
−Kii0 (~x)

=
+ Dii0 (~x)
∂t
∂ξi
2
∂ξi ∂ξi0
0
(A.18)
i,i =1
which agrees with Eq. (4.9).
A.3. Equations for hξi (t)i and σii (t) = hξiξi i
0
0
With the help of the Fokker-Planck equation (A.18)
one can derive equations for the
temporal evolution of the average of a function f ξ~ according to the scheme
D E
d f ξ~
dt
d
dt
Z
∞
Z
∞
~ t
dξk f ξ~ Π ξ,
(A.19)
−∞
−∞
Z ∞
~ t
∂Π ξ,
k
~
= Πi=1
dξi f ξ
∂t
−∞
h
i


2 Π ξ,
Z
~
~ t
k
0
∂
ξ
Π
ξ,
t
∂
X
j
1
(A.20)
=
dξ~ f ξ~ −Kjj 0
+ Djj 0
∂ξj
2
∂ξj ∂ξj 0
0
=
dξ1 · . . . ·
j,j =1
where we have introduced the notation
Z ∞
Z
Z
~
dξ ≡
dξ1 · . . . ·
−∞
∞
−∞
40
dξk =
Πki=1
Z
∞
dξi .
−∞
A. Derivation of the Linear Noise Approximation
Also, in order to rewrite the expressions in Eq. (A.20) in terms of average quantities we
will have to do partial integrations of the form
Z
Z
∂ ∂ h i
~
~
~ t
~
~
f ξ~
g ξ Π ξ, t
= − dξ~ g ξ~ Π ξ,
dξ f ξ
∂ξi
∂ξi
~ t , being a multivariate Gaussian (Eq.
where the boundary term vanishes since Π ξ,
4.11), decays exponentially as ξ~ → ∞.
For the mean value we obtain

h
i

2 Π ξ,
~ t
~ t
k Z
0 Π ξ,
∂
∂
ξ
X
j
d hξi i
1

=
+ Djj 0
dξ~ ξi −Kjj 0
dt
∂ξj
2
∂ξj ∂ξj 0
0
j,j =1
=
k Z
X
j,j 0 =1
=
k Z
X
dξ~
∂ξ
∂2ξ 1
i
i
~
~ t
+ Djj 0 Π ξ,
Kjj 0 ξj 0 Π ξ, t
∂ξj
2
∂ξj ∂ξj 0
~ t δij
dξ~ Kjj 0 ξj 0 Π ξ,
j,j 0 =1
=
k
X
Kij 0 ξj 0
,j 0 =1
or, in vector notation,
D E
d ξ~
dt
D E
= K · ξ~ ,
(A.21)
i.e. the uctuations ξ~ obey the same equation as small deviations from a steady state of
the mean-eld equations (Eq. A.15). Hence, if K (~x) is evaluated at an asymptotically
stable steady state all eigenvalues of K have a negative real part so that the uctuations
decay to zero in time.
41
A. Derivation of the Linear Noise Approximation
The calculation for the variance-covariance matrix σii0 = hξi ξi0 i yields

h
i

2 Π ξ,
~ t
~
k Z
0
∂
∂
ξ
Π
ξ,
t
X
j
1
d hξi ξi0 i
~


+ Djj 0
=
dξ ξi ξi0 −Kjj 0
0
dt
∂ξ
2
∂ξ
∂ξ
j
j
j
0
j,j =1
=
k Z
X
j,j 0 =1
=
k Z
X
dξ~
∂ 2 (ξ ξ 0 ) ∂ (ξ ξ 0 ) 1
i i
i i
~ t
~
+ Djj 0 Π ξ,
Kjj 0 ξj 0 Π ξ, t
∂ξj
2
∂ξj ∂ξj 0
~ t δij ξi0 + δi0 j ξi
dξ~ Kjj 0 ξj 0 Π ξ,
j,j 0 =1
+
k Z
1 X
~ t δij δi0 j 0 + δi0 j δij 0
dξ~ Djj 0 Π ξ,
2 0
j,j =1
=
k
X
j 0 =1
k
X
1
Kij 0 ξj 0 ξi0 +
Ki0 j 0 ξj 0 ξi + (Dii0 + Di0 i )
2
0
j =1
or, in matrix notation,
dσ
= K · σ + σ · KT + D
dt
where we have used the fact that D is symmetric (Dii0 = Di0 i ).
42
(A.22)
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43